Concept

Bernoulli convolution

The distribution of the random sum ±1 ± λ ± λ² ± …, with a fair coin for every sign. Below λ = 1/2 it lives on a Cantor set and at 1/2 it is flat; above 1/2 it has a density for almost every λ, and none when 1/λ is a Pisot number such as the golden ratio.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Where a coin-signed geometric series lands, for λ = 1/3, 1/2, 1/√2, 1/φ. Histograms of the exact distribution of the sum of plus or minus λ to the k, one panel per value of λ: a dust of separated pieces below one half, a flat block at one half, and smooth-looking overlapping shapes above.

A coin in front of every power

Toss a coin for each sign of ±1 ± λ ± λ² ± … and the sum lands somewhere. Below λ = 1/2 it lands on a dust with gaps in it, at exactly 1/2 it lands anywhere with equal chance, and above 1/2 it lands on a smooth-looking hill — which for most λ has a density and for the golden value does not, though no picture can tell the two apart.

analysis · Harmonic series

Sign patterns, and the far fewer points they land on. A logarithmic plot of the number of sign patterns, two to the n, against the number of distinct values the golden geometric sum takes, which is a Fibonacci number less one and falls further behind at every step.

Two sign patterns that land together

At λ = 1/φ the sign patterns + − − and − + + land in exactly the same place, because λ² + λ = 1. That one coincidence, repeated wherever it fits, puts 2ⁿ patterns onto a Fibonacci number of points, leaves the random sum's transform ringing at the same height forever, and makes a distribution that fills a whole interval live on a set of no length.

analysis · Harmonic series

Named alongside it

The objects these essays reach for when they reach for this one.

Fractal dimension Probability density Algebraic integer Almost surely Binary expansion Cantor set Characteristic function Entropy Fibonacci Geometric series Golden ratio Self-similarity

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